Let $X_1,X_2...X_n $ be a random sample from a $N(\theta ,\theta^2 )$ distribution, where $\theta >0$ is unknown.Let $T_1=\sum_{i=1}^{n}X_i$ and $T_2=\sum_{i=1}^{n}X_i^2$
Which of the following statements are correct?
$(A)=\dfrac{T_1^2}{n^2} $ is unbiased for $\theta^2$
$(B)=\dfrac{T_2}{2n} $ is unbiased for $\theta^2$
$(C)=\dfrac{T_1^2}{n^2} $ is consistent for $\theta^2$
$(D)=\dfrac{T_2}{2n} $ is consistent for $\theta^2$
The way I tried
$(A)\implies \dfrac{T_1}{n}=\bar X $ squaring both sides $\dfrac{T_1^2}{n^2}=\bar X^2$
If I take expectation I get $E(\bar X^2)=\theta^2$
$(B)\implies E(T_2)=\sum_{i=1}^{n}E(X_i^2)=n(V(X)+E(X)^2)=n(2\theta^2)$
$E\bigg(\dfrac{T_2}{2n}\bigg)=\theta^2$
Although this question has been answered here. But in option (A) I am making some mistake and I am not able to figure out.